systems_linear_equations - Math 568 Systems of Linear...

Math 568Systems of Linear Equations and MatricesIn these notes, we define a linear system and their associated matrices. We also indicate the algebra whichcan be preformed on these objects.1.Definitions and NotationAlinear equation innvariablesis an equation of the form:a1x1+a2x2+· · ·+anxn=band asystem ofmlinear equations innvariablesis a collection of linear equations in the samenvariables:a11x1+a12x2+· · ·+a1nxn=b1a21x1+a22x2+· · ·+a2nxn=b2...am1x1+am2x2+· · ·+amnxn=bmAsolutionto a system of linear equations innvariables is an vector [s1, s2, . . . , sn] such that the componentssatisfyallof the equations in the system when we setxi=si.We say that a system of linear equations isconsistentif it has at least one solution; otherwise we say that it isinconsistent. A system of linear equationsmay have more than one solution (we will see later that it must have infinitely many solutions in this case) andthe collection of all solutions of a linear system is called itssolution set.Consider the following two linear systems:x1+x2= 2x1-x2= 0andx1= 1x2= 1Notice that they have exactly the same solution set, namely{[1,1]}.We say that these two systems areequivalent. More generally, two systems of linear equations (in the same variables) are said to be

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